# American Institute of Mathematical Sciences

June  2013, 6(3): 687-701. doi: 10.3934/dcdss.2013.6.687

## Null controllability of degenerate parabolic equations in non divergence form via Carleman estimates

 1 Dipartimento di Matematica, Università di Bari, Via E. Orabona 4, 70125 Bari

Received  April 2010 Revised  April 2011 Published  December 2012

We prove null controllability results for the one dimensional degenerate heat equation in non divergence form with a drift term and Neumann boundary conditions. To this aim we prove Carleman estimates for the associated adjoint problem. Some linear extensions are considered.
Citation: Genni Fragnelli. Null controllability of degenerate parabolic equations in non divergence form via Carleman estimates. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 687-701. doi: 10.3934/dcdss.2013.6.687
##### References:
 [1] F. Alabau-Boussouira, P. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., 6 (2006), 161-204. doi: 10.1007/s00028-006-0222-6.  Google Scholar [2] A. Bensoussan, G. Da Prato, M. C. Delfout and S. K. Mitter, "Representation and Control of Infinite Dimensional Systems," Systems and Control: Foundations and Applications, Birkhäuser, Boston, 1993. Google Scholar [3] V. Barbu, A. Favini and S. Romanelli, Degenerate evolution equations and regularity of their associated semigroups, Funkc. Eqv., 39 (1996), 421-448.  Google Scholar [4] P. Cannarsa and G. Fragnelli, Null controllability of semilinear weakly degenerate parabolic equations in bounded domains, Electron. J. Differential Equations, 2006 (2006), 1-20.  Google Scholar [5] P. Cannarsa, G. Fragnelli and D. Rocchetti, Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form, J. Evol. Equ., 8 (2008), 583-616. doi: 10.1007/s00028-008-0353-34.  Google Scholar [6] P. Cannarsa, G. Fragnelli and D. Rocchetti, Null controllability of degenerate parabolic operators with drift, Netw. Heterog. Media, 2 (2007), 693-713. doi: 10.3934/nhm.2007.2.695.  Google Scholar [7] P. Cannarsa, G. Fragnelli and J. Vancostenoble, Linear degenerate parabolic equations in bounded domains: controllability and observability, IFIP Int. Fed. Inf. Process, 202 (2006), 163-173. Springer, New York. doi: 10.1007/0-387-33882-9_15.  Google Scholar [8] P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., 47 (2008), 1-19. doi: 10.1137/04062062X.  Google Scholar [9] P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates and null controllability for boundary-degenerate parabolic operators, C. R. Acad. Sci. Paris, Ser. I, 347 (2009), 147-152. doi: 10.1016/j.crma.2008.12.011.  Google Scholar [10] E. B. Davies, "Spectral Theory and Differential Operators," Cambridge Studies in Advanced Mathematics, 42. Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511623721.  Google Scholar [11] K. J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Springer-Verlag, New York, Berlin, Heidelberg, 1999.  Google Scholar [12] H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal., 43 (1971), 272-292.  Google Scholar [13] H. O. Fattorini, "Infinite Dimensional Optimization and Control Theory," Encyclopedia of Mathematics and its Applications, 62. Cambridge University Press, Cambridge, 1999.  Google Scholar [14] A. Favini, J. A. Goldstein and S. Romanelli, Analytic Semigroups on $L^p_\omega (0,1)$ Generated by Some Classes of Second Order Differential Operators, Taiwanese J. Math., 3 (1999), 181-210.  Google Scholar [15] A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Degenerate second order differential operators generating analytic semigroups in $L^p$ and $W^{1,p}$, Math. Nachr., 238 (2002), 78-102.  Google Scholar [16] A. Favini and A. Yagi, "Degenerate Differential Equations in Banach Spaces," Monographs and Textbooks in Pure and Applied Mathematics, 215. Marcel Dekker, Inc., New York, 1999.  Google Scholar [17] W. Feller, The parabolic differential equations and the associated semigroups of transformations, Ann. of Math. (2), 55 (1952), 468-519.  Google Scholar [18] W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc., 77 (1954), 1-31.  Google Scholar [19] E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 583-616. doi: 10.1016/S0294-1449(00)00117-7.  Google Scholar [20] A. V. Fursikov and O. Yu. Imanuvilov, "Controllability of Evolution Equations," Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.  Google Scholar [21] J. A. Goldstein, R. M. Mininni and S. Romanelli, A new explicit formula for the solution of the Black-Merton-Scholes equation, Infinite Dimensional Stochastic Analysis. Quantum Probab. White Noise Anal., World Sci. Publ., Hackensack, NJ, 22 (2008), 226-235. doi: 10.1142/9789812779557_0013.  Google Scholar [22] S. Karlin and H. M. Taylor, "A Second Course in Stochastic Processes," Academic Press, Inc. (Harcourt Brace Jovanovich, Publishers), New York-London, 1981.  Google Scholar [23] G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356. doi: 10.1080/03605309508821097.  Google Scholar [24] P. Mandl, "Analytical Treatment of One-Dimensional Markov Processes," Die Grundlehren der mathematischen Wissenschaften, 151 Academia Publishing House of the Czechoslovak Academy of Sciences, Prague; Springer-Verlag New York Inc., New York, 1968.  Google Scholar [25] P. Martinez and J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Equ., 6 (2006), 325-362. doi: 10.1007/s00028-006-0214-6.  Google Scholar [26] P. Martinez, J. P. Raymond and J. Vancostenoble, Regional null controllability of a linearized Crocco-type equation, SIAM J. Control Optim., 42 (2003), 709-728. doi: 10.1137/S0363012902403547.  Google Scholar [27] G. Metafune and D. Pallara, Trace formulas for some singular differential operators and applications, Math. Nachr., 211 (2000), 127-157.  Google Scholar [28] S. Micu and E. Zuazua, On the lack of null controllability of the heat equation on the half-space, Port. Math. (N.S.), 58 (2001), 1-24.  Google Scholar [29] S. Micu and E. Zuazua, On the lack of null controllability of the heat equation on the half-line, Trans. Amer. Math. Soc., 353 (2001), 1635-1659. doi: 10.1090/S0002-9947-00-02665-9.  Google Scholar [30] R. M. Mininni and S. Romanelli, Martingale estimating functions for Feller diffusion processes generated by degenerate elliptic operators, J. Concr. Appl. Math., 1 (2003), 191-216.  Google Scholar [31] D. L. Russell, Controllability and stabilizability theorems for linear partial differential equations: recent progress and open questions, SIAM Rev., 20 (1978), 639-739. doi: 10.1137/1020095.  Google Scholar [32] N. Shimakura, "Partial Differential Operators of Elliptic Type," Translations of Mathematical Monographs, 99. American Mathematical Society, Providence, RI, 1992.  Google Scholar [33] D. Tataru, Carleman estimates, unique continuation and controllability for anizotropic PDE's, Contemp. Math., 209 (1997), 267-279. doi: 10.1090/conm/209/02771.  Google Scholar

show all references

##### References:
 [1] F. Alabau-Boussouira, P. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., 6 (2006), 161-204. doi: 10.1007/s00028-006-0222-6.  Google Scholar [2] A. Bensoussan, G. Da Prato, M. C. Delfout and S. K. Mitter, "Representation and Control of Infinite Dimensional Systems," Systems and Control: Foundations and Applications, Birkhäuser, Boston, 1993. Google Scholar [3] V. Barbu, A. Favini and S. Romanelli, Degenerate evolution equations and regularity of their associated semigroups, Funkc. Eqv., 39 (1996), 421-448.  Google Scholar [4] P. Cannarsa and G. Fragnelli, Null controllability of semilinear weakly degenerate parabolic equations in bounded domains, Electron. J. Differential Equations, 2006 (2006), 1-20.  Google Scholar [5] P. Cannarsa, G. Fragnelli and D. Rocchetti, Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form, J. Evol. Equ., 8 (2008), 583-616. doi: 10.1007/s00028-008-0353-34.  Google Scholar [6] P. Cannarsa, G. Fragnelli and D. Rocchetti, Null controllability of degenerate parabolic operators with drift, Netw. Heterog. Media, 2 (2007), 693-713. doi: 10.3934/nhm.2007.2.695.  Google Scholar [7] P. Cannarsa, G. Fragnelli and J. Vancostenoble, Linear degenerate parabolic equations in bounded domains: controllability and observability, IFIP Int. Fed. Inf. Process, 202 (2006), 163-173. Springer, New York. doi: 10.1007/0-387-33882-9_15.  Google Scholar [8] P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., 47 (2008), 1-19. doi: 10.1137/04062062X.  Google Scholar [9] P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates and null controllability for boundary-degenerate parabolic operators, C. R. Acad. Sci. Paris, Ser. I, 347 (2009), 147-152. doi: 10.1016/j.crma.2008.12.011.  Google Scholar [10] E. B. Davies, "Spectral Theory and Differential Operators," Cambridge Studies in Advanced Mathematics, 42. Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511623721.  Google Scholar [11] K. J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Springer-Verlag, New York, Berlin, Heidelberg, 1999.  Google Scholar [12] H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal., 43 (1971), 272-292.  Google Scholar [13] H. O. Fattorini, "Infinite Dimensional Optimization and Control Theory," Encyclopedia of Mathematics and its Applications, 62. Cambridge University Press, Cambridge, 1999.  Google Scholar [14] A. Favini, J. A. Goldstein and S. Romanelli, Analytic Semigroups on $L^p_\omega (0,1)$ Generated by Some Classes of Second Order Differential Operators, Taiwanese J. Math., 3 (1999), 181-210.  Google Scholar [15] A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Degenerate second order differential operators generating analytic semigroups in $L^p$ and $W^{1,p}$, Math. Nachr., 238 (2002), 78-102.  Google Scholar [16] A. Favini and A. Yagi, "Degenerate Differential Equations in Banach Spaces," Monographs and Textbooks in Pure and Applied Mathematics, 215. Marcel Dekker, Inc., New York, 1999.  Google Scholar [17] W. Feller, The parabolic differential equations and the associated semigroups of transformations, Ann. of Math. (2), 55 (1952), 468-519.  Google Scholar [18] W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc., 77 (1954), 1-31.  Google Scholar [19] E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 583-616. doi: 10.1016/S0294-1449(00)00117-7.  Google Scholar [20] A. V. Fursikov and O. Yu. Imanuvilov, "Controllability of Evolution Equations," Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.  Google Scholar [21] J. A. Goldstein, R. M. Mininni and S. Romanelli, A new explicit formula for the solution of the Black-Merton-Scholes equation, Infinite Dimensional Stochastic Analysis. Quantum Probab. White Noise Anal., World Sci. Publ., Hackensack, NJ, 22 (2008), 226-235. doi: 10.1142/9789812779557_0013.  Google Scholar [22] S. Karlin and H. M. Taylor, "A Second Course in Stochastic Processes," Academic Press, Inc. (Harcourt Brace Jovanovich, Publishers), New York-London, 1981.  Google Scholar [23] G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356. doi: 10.1080/03605309508821097.  Google Scholar [24] P. Mandl, "Analytical Treatment of One-Dimensional Markov Processes," Die Grundlehren der mathematischen Wissenschaften, 151 Academia Publishing House of the Czechoslovak Academy of Sciences, Prague; Springer-Verlag New York Inc., New York, 1968.  Google Scholar [25] P. Martinez and J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Equ., 6 (2006), 325-362. doi: 10.1007/s00028-006-0214-6.  Google Scholar [26] P. Martinez, J. P. Raymond and J. Vancostenoble, Regional null controllability of a linearized Crocco-type equation, SIAM J. Control Optim., 42 (2003), 709-728. doi: 10.1137/S0363012902403547.  Google Scholar [27] G. Metafune and D. Pallara, Trace formulas for some singular differential operators and applications, Math. Nachr., 211 (2000), 127-157.  Google Scholar [28] S. Micu and E. Zuazua, On the lack of null controllability of the heat equation on the half-space, Port. Math. (N.S.), 58 (2001), 1-24.  Google Scholar [29] S. Micu and E. Zuazua, On the lack of null controllability of the heat equation on the half-line, Trans. Amer. Math. Soc., 353 (2001), 1635-1659. doi: 10.1090/S0002-9947-00-02665-9.  Google Scholar [30] R. M. Mininni and S. Romanelli, Martingale estimating functions for Feller diffusion processes generated by degenerate elliptic operators, J. Concr. Appl. Math., 1 (2003), 191-216.  Google Scholar [31] D. L. Russell, Controllability and stabilizability theorems for linear partial differential equations: recent progress and open questions, SIAM Rev., 20 (1978), 639-739. doi: 10.1137/1020095.  Google Scholar [32] N. Shimakura, "Partial Differential Operators of Elliptic Type," Translations of Mathematical Monographs, 99. American Mathematical Society, Providence, RI, 1992.  Google Scholar [33] D. Tataru, Carleman estimates, unique continuation and controllability for anizotropic PDE's, Contemp. Math., 209 (1997), 267-279. doi: 10.1090/conm/209/02771.  Google Scholar
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